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A115720 Triangle T(n,k) is the number of partitions of n with Durfee square k.

%I A115720 #22 Feb 16 2025 08:33:00
%S A115720 1,0,1,0,2,0,3,0,4,1,0,5,2,0,6,5,0,7,8,0,8,14,0,9,20,1,0,10,30,2,0,11,
%T A115720 40,5,0,12,55,10,0,13,70,18,0,14,91,30,0,15,112,49,0,16,140,74,1,0,17,
%U A115720 168,110,2,0,18,204,158,5,0,19,240,221,10,0,20,285,302,20,0,21,330,407
%N A115720 Triangle T(n,k) is the number of partitions of n with Durfee square k.
%C A115720 T(n,k) is number of partitions of n-k^2 into parts of 2 kinds with at most k of each kind.
%H A115720 Alois P. Heinz, <a href="/A115720/b115720.txt">Rows n = 0..600, flattened</a>
%H A115720 Findstat, <a href="http://www.findstat.org/StatisticsDatabase/St000183/">St000183: The side length of the Durfee square of an integer partition</a>
%H A115720 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DurfeeSquare.html">Durfee Square</a>
%F A115720 T(n,k) = Sum_{i=0..n-k^2} P*(i,k)*P*(n-k^2-i), where P*(n,k) = P(n+k,k) is the number of partitions of n objects into at most k parts.
%e A115720 Triangle starts:
%e A115720   1;
%e A115720   0,  1;
%e A115720   0,  2;
%e A115720   0,  3;
%e A115720   0,  4,  1;
%e A115720   0,  5,  2;
%e A115720   0,  6,  5;
%e A115720   0,  7,  8;
%e A115720   0,  8, 14;
%e A115720   0,  9, 20,  1;
%e A115720   0, 10, 30,  2;
%e A115720 From _Gus Wiseman_, Apr 12 2019: (Start)
%e A115720 Row n = 9 counts the following partitions:
%e A115720   (9)          (54)       (333)
%e A115720   (81)         (63)
%e A115720   (711)        (72)
%e A115720   (6111)       (432)
%e A115720   (51111)      (441)
%e A115720   (411111)     (522)
%e A115720   (3111111)    (531)
%e A115720   (21111111)   (621)
%e A115720   (111111111)  (3222)
%e A115720                (3321)
%e A115720                (4221)
%e A115720                (4311)
%e A115720                (5211)
%e A115720                (22221)
%e A115720                (32211)
%e A115720                (33111)
%e A115720                (42111)
%e A115720                (222111)
%e A115720                (321111)
%e A115720                (2211111)
%e A115720 (End)
%p A115720 b:= proc(n, i) option remember;
%p A115720       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p A115720     end:
%p A115720 T:= (n, k)-> add(b(m, k)*b(n-k^2-m, k), m=0..n-k^2):
%p A115720 seq(seq(T(n, k), k=0..floor(sqrt(n))), n=0..30); # _Alois P. Heinz_, Apr 09 2012
%t A115720 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; T[n_, k_] := Sum[b[m, k]*b[n-k^2-m, k], {m, 0, n-k^2}]; Table[ T[n, k], {n, 0, 30}, {k, 0, Sqrt[n]}] // Flatten (* _Jean-François Alcover_, Dec 03 2015, after _Alois P. Heinz_ *)
%t A115720 durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
%t A115720 Table[Length[Select[IntegerPartitions[n],durf[#]==k&]],{n,0,10},{k,0,Sqrt[n]}] (* _Gus Wiseman_, Apr 12 2019 *)
%Y A115720 For a version without zeros see A115994. Row lengths are A003059. Row sums are A000041. Column k = 2 is A006918. Column k = 3 is A117485.
%Y A115720 Cf. A008284, A115721, A115722, A257990, A325164.
%Y A115720 Related triangles are A096771, A325188, A325189, A325192, with Heinz-encoded versions A263297, A325169, A065770, A325178.
%K A115720 nonn,tabf
%O A115720 0,5
%A A115720 _Franklin T. Adams-Watters_, Mar 11 2006